Example. Let RRR be a ring. The category of left (right) RRR-modules is an abelian category having enough projectives. This is true since, for every left (right) RRR-module MMM, we can take FFF to be the free (and hence projective) RRR-module generated by a generating setXXX for MMM (we can in fact take XXX to be MMM). Then the canonical projectionπ:F→Mnormal-:πnormal-→FM\pi\colon F\to M is the required surjection.

More generally, a category ??\mathcal{C} is said to have enough projectives if every object is a strongquotient object of a projective object.